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1 History and Perspective1

1.1 Brief History of the Science of Electromagnetism1

1.2 Electromagnetism in the Standard Model5

2 Vector Calculus9

2.1 Vector Algebra10

2.1.1 Definitions10

2.1.2 Addition and Multiplication of Vectors13

2.1.3 Vector Product Identities14

2.1.4 Geometric Meanings16

2.2 Vector Differential Operators18

2.2.1 Gradient of a Scalar Function18

2.2.2 Divergence of a Vector Function19

2.2.3 Curl of a Vector Function20

2.2.4 DelIdentities23

2.3 Integral Theorems25

2.3.1 Gauss’s Theorem26

2.3.2 Stokes’s Theorem27

2.3.3 Vector Calculus in Fluid Mechanics29

2.4 Curvilinear Coordinates30

2.4.1 General Derivations30

2.4.2 Cartesian, Cylindrical, and Spherical Coordinates33

2.5 The Helmholtz Theorem37

3 Basic Principles of Electrostatics44

3.1 Coulomb’s Law44

3.1.1 The Superposition Principle46

3.2 The Electric Field46

3.2.1 Definition46

3.2.2 Charge as the Source of E47

3.2.3 Field of a Charge Continuum49

3.3 Curl and Divergence of E54

3.3.1 Field Theory Versus Action at a Distance56

3.3.2 Boundary Conditions of the Electrostatic Field56

3.4 The Integral Form of Gauss’s Law57

3.4.1 Flux and Charge57

3.4.2 Proof of Gauss’s Law57

3.4.3 Calculations Based on Gauss’s Law59

3.5 Green’s Function and the Dirac delta Function62

3.5.1 The Dirac delta Function62

3.5.2 Another Proof of Gauss’s Law65

3.6 The Electric Potential65

3.6.1 Definition and Construction65

3.6.2 Poisson’s Equation68

3.6.3 Example Calculations of V （x）69

3.7 Energy of the Electric Field72

3.8 The Multipole Expansion75

3.8.1 Two Charges75

3.8.2 The Electric Dipole77

3.8.3 Moments of a General Charge Distribution78

3.8.4 Equipotentials and Field Lines79

3.8.5 Torque and Potential Energy for a Dipole in an Electric Field80

3.9 Applications82

3.10 Chapter Summary83

4 Electrostatics and Conductors92

4.1 Electrostatic properties of conductors93

4.2 Electrostatic Problems with Rectangular Symmetry98

4.2.1 Charged Plates98

4.2.2 Problems with Rectangular Symmetry and External Point Charges.The Method of Images102

4.3 Problems with Spherical Symmetry107

4.3.1 Charged Spheres107

4.3.2 Problems with Spherical Symmetry and External Charges113

4.4 Problems with Cylindrical Symmetry116

4.4.1 Charged Lines and Cylinders116

4.4.2 Problems with Cylindrical Symmetry and an External Line Charge124

5 General Methods for Laplace’s Equation133

5.1 Separation of Variables for Cartesian Coordinates135

5.1.1 Separable Solutions for Cartesian Coordinates136

5.1.2 Examples138

5.2 Separation of Variables for Spherical Polar Coordinates147

5.2.1 Separable Solutions for Spherical Coordinates147

5.2.2 Legendre Polynomials149

5.2.3 Examples with Spherical Boundaries150

5.3 Separation of Variables for Cylindrical Coordinates159

5.3.1 Separable Solutions for Cylindrical Coordinates160

5.4 Conjugate Functions in 2 Dimensions163

5.5 Iterative Relaxation: A Numerical Method172

6 Electrostatics and Dielectrics186

6.1 The Atom as an Electric Dipole187

6.1.1 Induced Dipoles187

6.1.2 Polar Molecules189

6.2 Polarization and Bound Charge191

6.3 The Displacement Field195

6.3.1 Linear Dielectrics197

6.3.2 The Clausius-Mossotti Formula198

6.3.3 Poisson’s Equation in a Uniform Linear Dielectric200

6.4 Dielectric Material in a Capacitor201

6.4.1 Design of Capacitors203

6.4.2 Microscopic Theory204

6.4.3 Energy in a Capacitor205

6.4.4 A Concrete Model of a Dielectric207

6.5 Boundary Value Problems with Dielectrics208

6.5.1 The Boundary Conditions208

6.5.2 A Dielectric Sphere in an Applied Field209

6.5.3 A Point Charge above a Dielectric with a Planar Bound-ary Surface211

6.5.4 A Capacitor Partially Filled with Dielectric212

7 Electric Currents222

7.1 Electric Current in a Wire222

7.2 Current Density and the Continuity Equation224

7.2.1 Local Conservation of Charge226

7.2.2 Boundary Condition on J（x, t）226

7.3 Current and Resistance228

7.3.1 Ohm’s Law228

7.3.2 Fabrication of Resistors233

7.3.3 The Surface Charge on a Current Carrying Wire234

7.4 A Classical Model of Conductivity236

7.5 Joule’s Law238

7.6 Decay of a Charge Density Fluctuation239

7.7 Ⅰ-Ⅴ Characteristic of a Vacuum-Tube Diode241

7.8 Chapter Summary246

8 Magnetostatics252

8.1 The Magnetic Force and the Magnetic Field253

8.1.1 Force on a Moving Charge253

8.1.2 Force on a Current-Carrying Wire255

8.2 Applications of the Magnetic Force255

8.2.1 Helical or Circular Motion of q in Uniform B255

8.2.2 Cycloidal Motion of q in Crossed E and B258

8.2.3 Electric Motors260

8.3 Electric Current as a Source of Magnetic Field262

8.3.1 The Biot-Savart Law262

8.3.2 Forces on Parallel Wires266

8.3.3 General Field Equations for B（x）267

8.4 Ampere’s Law270

8.4.1 Ampere Law Calculations271

8.4.2 Formal Proof of Ampere’s Law277

8.5 The Vector Potential280

8.5.1 General Solution for A（x）281

8.6 The Magnetic Dipole284

8.6.1 Asymptotic Analysis284

8.6.2 Dipole Moment of a Planar Loop286

8.6.3 Torque and Potential Energy of a Magnetic Dipole287

8.6.4 The Magnetic Field of the Earth291

8.7 The Full Field of a Current Loop291

9 Magnetic Fields and Matter307

9.1 The Atom as a Magnetic Dipole307

9.1.1 Diamagnetism310

9.1.2 Paramagnetism313

9.2 Magnetization and Bound Currents314

9.2.1 Examples316

9.2.2 A Geometric Derivation of the Bound Currents320

9.3 Ampere’s Law for Free Currents, and H323

9.3.1 The Integral Form of Ampere’s Law326

9.3.2 The Constitutive Equation326

9.3.3 Magnetic Susceptibilities326

9.3.4 Boundary Conditions for Magnetic Fields329

9.4 Problems Involving Free Currents and Magnetic Materials331

9.5 A Magnetic Body in an External Field: The Magnetic Scalar Potentialφm （x）335

9.6 Ferromagnetism342

9.6.1 Measuring Magnetization Curves: The Rowland Ring343

9.6.2 Magnetization Curves of Ferromagnetic Materials345

9.6.3 The Permeability of a Ferromagnetic Material346

10 Electromagnetic Induction355

10.1 Motional EMF356

10.1.1 Electromotive Force356

10.1.2 EMF from Motion in B357

10.1.3 The Faraday Disk Generator358

10.2 Faraday’s Law of Electromagnetic Induction360

10.2.1 Mathematical Statement361

10.2.2 Lenz’s Law363

10.2.3 Eddy Currents364

10.3 Applications of Faraday’s Law368

10.3.1 The Electric Generator and Induction Motor369

10.3.2 The Betatron371

10.3.3 Self-Inductance372

10.3.4 Classical Model of Diamagnetism375

10.4 Mutual Inductance376

10.5 Magnetic Field Energy382

10.5.1 Energy in a Ferromagnet386

11 The Maxwell Equations397

11.1 The Maxwell Equations in Vacuum and the Displacement Current398

11.1.1 The Displacement Current399

11.2 Scalar and Vector Potentials405

11.2.1 Gauge Transformations and Gauge Invariance406

11.2.2 Gauge Choices and Equations for A（x,t） and V（x,t）407

11.3 The Maxwell Equations in Matter410

11.3.1 Free and Bound Charge and Current410

11.3.2 Boundary Conditions of Fields413

11.4 Energy and Momentum of Electromagnetic Fields415

11.4.1 Poynting’s Theorem416

11.4.2 Field Momentum421

11.5 Electromagnetic Waves in Vacuum423

11.5.1 Derivation of the Wave Equation424

11.5.2 An Example of a Plane Wave Solution425

11.5.3 Derivation of the General Plane Wave Solution431

11.5.4 A Spherical Harmonic Wave434

11.5.5 The Theory of Light437

12 Electromagnetism and Relativity445

12.1 Coordinate Transformations446

12.1.1 The Galilean Transformation446

12.1.2 The Lorentz Transformation448

12.1.3 Examples Involving the Lorentz Transformation450

12.2 Minkowski Space452

12.2.1 4-vectors, Scalars, and Tensors452

12.2.2 Kinematics of a Point Particle455

12.2.3 Relativistic Dynamics457

12.3 Electromagnetism in Covariant Form458

12.3.1 The Lorentz Force and the Field Tensor458

12.3.2 Maxwell’s Equations in Covariant Form460

12.3.3 The 4-vector Potential462

12.4 Field Transformations463

12.5 Fields Due to a Point Charge in Uniform Motion468

12.6 Magnetism from Relativity474

12.7 The Energy-Momentum Flux Tensor477

13 Electromagnetism and Optics485

13.1 Electromagnetic Waves in a Dielectric485

13.2 Reflection and Refraction at a Dielectric Interface488

13.2.1 Wave Vectors490

13.2.2 Reflectivity for Normal Incidence494

13.2.3 Reflection for Incidence at Arbitrary Angles: Fresnel’s Equations498

13.3 Electromagnetic Waves in a Conductor505

13.3.1 Reflectivity of a Good Conductor509

13.4 A Classical Model of Dispersion: The Frequency Dependence of Material Properties511

13.4.1 Dispersion in a Dielectric512

13.4.2 Dispersion in a Plasma514

14 Wave Guides and Transmission Lines523

14.1 Electromagnetic Waves Between Parallel Conducting Planes524

14.1.1 The TEM Solution526

14.1.2 TE Waves528

14.1.3 TM Waves537

14.1.4 Summary540

14.2 The Rectangular Wave Guide540

14.2.1 Transverse Electric Modes TE（m, n）541

14.2.2 Transverse Magnetic Modes TM（m, n）547

14.3 Wave Guide of Arbitrary Shape549

14.4 The TEM Mode of a Coaxial Cable551

14.5 Cavity Resonance555

15 Radiation of Electromagnetic Waves560

15.1 The Retarded Potentials561

15.1.1 Green’s Functions561

15.2 Radiation from an Electric Dipole567

15.2.1 The Hertzian Dipole571

15.2.2 Atomic Transitions574

15.2.3 Magnetic Dipole Radiation575

15.2.4 Complete Fields of a Hertzian Dipole577

15.3 The Half-Wave Linear Antenna579

15.4 The Larmor Formula: Radiation from a Point Charge584

15.5 Classical Electron Theory of Light Scattering589

15.6 Complete Fields of a Point Charge: The Lienard-Wiechert Potentials593

15.6.1 A Charge with Constant Velocity596

15.6.2 The Complete Fields598

15.6.3 Generalization of the Larmor Formula599

A Electric and Magnetic Units607

B The Helmholtz Theorem610

Index613